html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 KelvinBer

 http://functions.wolfram.com/03.18.06.0028.01

 Input Form

 KelvinBer[-2 n - 1, z] == (((-1)^Floor[(n + 1)/2] 2^(-(3/2) - 2 n) z^(1 + 2 n))/(2 n + 1)!) Sum[((-1)^k (z/4)^(4 k))/(Pochhammer[1/2, k] Pochhammer[3/2 + n, k] Pochhammer[1 + n, k] k!), {k, 0, Infinity}] + (((-1)^Floor[n/2] 2^(-(7/2) - 2 n) z^(3 + 2 n))/(2 n + 2)!) Sum[((-1)^k (z/4)^(4 k))/(Pochhammer[3/2, k] Pochhammer[3/2 + n, k] Pochhammer[2 + n, k] k!), {k, 0, Infinity}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], "n"]], "-", "1"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "+", "1"]], "2"], "]"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["7", "2"]]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "2"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", "n"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 ber - 2 n - 1 ( z ) ( - 1 ) n + 1 2 2 - 2 n - 3 2 z 2 n + 1 ( 2 n + 1 ) ! k = 0 ( - 1 ) k ( z 4 ) 4 k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( n + 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], "k"], Pochhammer] ( n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "k"], Pochhammer] k ! + ( - 1 ) n 2 2 - 2 n - 7 2 z 2 n + 3 ( 2 n + 2 ) ! k = 0 ( - 1 ) k ( z 4 ) 4 k ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ( n + 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], "k"], Pochhammer] ( n + 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], "k"], Pochhammer] k ! /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition KelvinBer -2 n -1 z -1 n 1 2 -1 2 -2 n -1 3 2 z 2 n 1 2 n 1 -1 k 0 -1 k z 4 -1 4 k Pochhammer 1 2 k Pochhammer n 3 2 k Pochhammer n 1 k k -1 -1 n 2 -1 2 -2 n -1 7 2 z 2 n 3 2 n 2 -1 k 0 -1 k z 4 -1 4 k Pochhammer 3 2 k Pochhammer n 3 2 k Pochhammer n 2 k k -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n_"]], "-", "1"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "+", "1"]], "2"], "]"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["7", "2"]]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02